When is hyponormality for 2-variable weighted shifts invariant under
powers?

By Raul Curto and Jasang Yoon

Abstract

For 2-variable weighted shifts W_{(\alpha,\beta)}(T_1, T_2) we study the
invariance of (joint) k- hyponormality under the action (h,\ell) ->
W_{(\alpha,\beta)}^{(h,\ell)}(T_1, T_2):=(T_1^k,T_2^{\ell}) (h,\ell >=1). We
show that for every k >= 1 there exists W_{(\alpha,\beta)}(T_1, T_2) such that
W_{(\alpha,\beta)}^{(h,\ell)}(T_1, T_2) is k-hyponormal (all h>=2,\ell>=1) but
W_{(\alpha,\beta)}(T_1, T_2) is not k-hyponormal. On the positive side, for a
class of 2-variable weighted shifts with tensor core we find a computable
necessary condition for invariance. Next, we exhibit a large nontrivial class
for which hyponormality is indeed invariant under all powers; moreover, for
this class 2-hyponormality automatically implies subnormality. Our results
partially depend on new formulas for the determinant of generalized Hilbert
matrices and on criteria for their positive semi-definiteness